There is not enough information here for us to really understand what is going on. The first thing to do is to go there in person and have him explain in detail what they did to the car and how much each section of work cost in labor and parts to add up to $10,735.00. Then, ask them why they performed the additional work without running it by you first. He may respond that it was not additional work but that it is cost overage on the $10,735.00. If so, that should be apparent from the detailed break down for his estimate as of 6/15 and his invoice as of 10/9.
You may want to locate a classic car restoration expert in your area and pay him a fee for an hour or two to accompany you there so that he can understand the lingo and translate for you if the numbers that Doug Jenkens throws out make any sense. See here. If push comes to shove, you will need such an expert to testify on your case if you decide to go to court; so it is best to get one on board now.
If you suspect dishonesty or unfair trade practices, then you can sue in small claims court for any overages or unauthorized charges. See here.
You can also file or threaten to file a complaint with the AG's office. See here.
In Missouri, state consumer laws prohibit unfair and deceptive practices in auto repair. Mechanics who mislead, deceive, or make misrepresentations to consumers may be subject to penalties under the Merchandising Practices Act found in Chapter 407 of the Missouri Revised Statutes. Attorney's fees are also recoverable so this should provide you leverage to get this resolved quickly. See law here. If you file suit under this statute you should hire a lawyer and it will most likely need to be filed in Circuit Court.
If you choose to sue in small claims court, you can sue for fraud, misrepresentation, negligence and /or breach of contract depending on what you find out.